量子观测量有效测量的Cartan子代数方法
Cartan sub-algebra approach to efficient measurements of quantum observables
摘要：
在当前可用的量子硬件上，对应于物理可观察量的任意算子不能在单次测量中测量。为了获得可观测的期望值，需要将其算子划分为可测量的片段。然而，可观测量和它的碎片通常不共享任何本征态，因此即使当所准备的波函数接近可观测量的本征态时，获得可观测量的期望值所需的测量次数也会迅速增长。我们提供了一个统一的李代数框架来发展量子可观测量的有效测量方案，它基于两个要素：1）将可观测算子嵌入到李代数中; 2）利用酉算子将李代数元素转换为Cartan子代数（CSA）的元素. CSA起着核心作用，因为它的所有元素都是相互交换的，因此可以同时测量。我们说明了测量的框架出现在量子化学的变分量子本征求解器方法的期望值的哈密顿量。CSA方法将许多最近提出的用于测量优化的方法放在单个框架内，并且不仅允许减少可测量片段的数量，而且允许减少测量的总数。
英文摘要：
An arbitrary operator corresponding to a physical observable cannot be measured in a single measurement on currently available quantum hardware. To obtain the expectation value of the observable, one needs to partition its operator to measurable fragments. However, the observable and its fragments generally do not share any eigenstates, and thus the number of measurements needed to obtain the expectation value of the observable can grow rapidly even when the wavefunction prepared is close to an eigenstate of the observable. We provide a unified Lie algebraic framework for developing efficient measurement schemes for quantum observables, it is based on two elements: 1) embedding the observable operator in a Lie algebra and 2) transforming Lie algebra elements into those of a Cartan sub-algebra (CSA) using unitary operators. The CSA plays the central role because all its elements are mutually commutative and thus can be measured simultaneously. We illustrate the framework on measuring expectation values of Hamiltonians appearing in the Variational Quantum Eigensolver approach to quantum chemistry. The CSA approach puts many recently proposed methods for the measurement optimization within a single framework, and allows one not only to reduce the number of measurable fragments but also the total number of measurements.
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